Move vertices same distance as a shape keyed vertex / add vertices to shape key
I can make 3 examples of 3 different ways of doing this (and there can be many more).

1 - Vertex parent: select 3 vertices of the eyelid, then in object mode select the eyelash; shift select the eyelid and press Ctrl P (choose vertex triangle), so that eyelashes will follow any movement of the 3 choosen vertices.

2 - Drivers: set a shape key for the eyelash also and set a driver so that it will act together with the eyelid shapekey.

3 - Join the two objects: select the eyelash FISRT, then shift select the eyelid, press ctrl J, then select the shapekey and in edit mode put the eyelash in the correct position.